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Improving the learning of numeracy through formative assessment

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Title: Improving the learning of numeracy through formative assessment


1
Improving the learning of numeracy through
formative assessment
  • Dylan Wiliam
  • National Numeracy Conference
  • Edinburgh, March 2009
  • www.dylanwiliam.net

2
Raising achievement matters
  • Which of the following categories of skill is
    disappearing from the work-place most rapidly?
  • Routine manual
  • Non-routine manual
  • Routine cognitive
  • Complex communication
  • Expert thinking/problem-solving

3
but what is learned matters too
Autor, Levy Murnane, 2003
4
The only 21st century skill
  • So the model that says learn while youre at
    school, while youre young, the skills that you
    will apply during your lifetime is no longer
    tenable. The skills that you can learn when
    youre at school will not be applicable. They
    will be obsolete by the time you get into the
    workplace and need them, except for one skill.
    The one really competitive skill is the skill of
    being able to learn. It is the skill of being
    able not to give the right answer to questions
    about what you were taught in school, but to make
    the right response to situations that are outside
    the scope of what you were taught in school. We
    need to produce people who know how to act when
    theyre faced with situations for which they were
    not specifically prepared.
  • (Papert, 1998)

5
Formative assessment
Assessment for learning is any assessment for
which the first priority in its design and
practice is to serve the purpose of promoting
pupils learning. It thus differs from assessment
designed primarily to serve the purposes of
accountability, or of ranking, or of certifying
competence. An assessment activity can help
learning if it provides information to be used as
feedback, by teachers, and by their pupils, in
assessing themselves and each other, to modify
the teaching and learning activities in which
they are engaged. Such assessment becomes
formative assessment when the evidence is
actually used to adapt the teaching work to meet
learning needs. (Black et al., 2002)
6
Types of formative assessment
  • Long-cycle
  • Span across units, terms
  • Length four weeks to one year
  • Medium-cycle
  • Span within and between teaching units
  • Length one to four weeks
  • Short-cycle
  • Span within and between lessons
  • Length
  • day-by-day 24 to 48 hours
  • minute-by-minute 5 seconds to 2 hours

7
Unpacking formative assessment
  • Key processes
  • Establishing where the learners are in their
    learning
  • Establishing where they are going
  • Working out how to get there
  • Participants
  • Teachers
  • Peers
  • Learners

8
Aspects of formative assessment
9
Five key strategies
  • Clarifying, understanding, and sharing learning
    intentions
  • curriculum philosophy (goals and horizons)
  • Engineering effective classroom discussions,
    tasks and activities that elicit evidence of
    learning
  • classroom discourse, interactive whole-class
    teaching
  • Providing feedback that moves learners forward
  • feedback
  • Activating students as learning resources for one
    another
  • collaborative learning, reciprocal teaching,
    peer-assessment
  • Activating students as owners of their own
    learning
  • metacognition, motivation, interest, attribution,
    self-assessment

(Wiliam Thompson, 2007)
10
and one big idea
  • Use evidence about learning to adapt teaching and
    learning to meet student needs

11
Keeping Learning on Track (KLT)
  • A pilot guides a plane or boat toward its
    destination by taking constant readings and
    making careful adjustments in response to wind,
    currents, weather, etc.
  • A good teacher does the same
  • Plans a carefully chosen route ahead of time (in
    essence building the track)
  • Takes readings along the way
  • Changes course as conditions dictate

12
Eliciting evidence of student achievement
13
Kinds of questions Israel
Which fraction is the smallest?
Success rate 88
Which fraction is the largest?
Success rate 46 39 chose (b)
Vinner, PME conference, Lahti, Finland, 1997
14
Draw an upside-down triangle
15
Misconceptions
3a 24 a b 16
16
Questioning in maths discussion
  • Look at the following sequence
  • 3, 7, 11, 15, 19, .
  • Which is the best rule to describe the sequence?
  • n 4
  • 3 n
  • 4n - 1
  • 4n 3

17
Eliciting evidence
  • Key idea questioning should
  • cause thinking
  • provide data that informs teaching
  • Improving teacher questioning
  • generating questions with colleagues
  • closed v open
  • low-order v high-order
  • appropriate wait-time
  • Getting away from I-R-E
  • basketball rather than serial table-tennis
  • No hands up (except to ask a question)
  • class polls to review current attitudes towards
    an issue
  • Hot Seat questioning
  • All-student response systems
  • ABCD cards, Mini white-boards, Exit passes

18
Questioning in maths diagnosis
  • In which of these right-angled triangles is a2
    b2 c2 ?

19
(No Transcript)
20
Lines of symmetry
C
A
B
F
D
E
21
Constructing hinge-point questions
22
Discriminate incorrect cognitive rules
  • Version 1 (Hart, 1981)
  • If ef  8, then efg  
  • 9
  • 12
  • 15
  • 8g
  • Version 2
  • If fg  8, then fgh  
  • 9
  • 12
  • 15
  • 16
  • 8h

23
Discriminate correct cognitive rules
What is the area of this trapezium?
a
h
b
24
2A (a  b) x h A  (a  b) x h
A  h x (a  b)
25
Discriminate between incorrect and correct
cognitive rules
  • Version 1
  • There are two flights per day from Newtown to
    Oldtown. The first flight leaves Newtown each day
    at 920 and arrives in Oldtown at 1055. The
    second flight from Newtown leaves at 215. At
    what time does the second flight arrive in
    Oldtown? Show your work.
  • Version 2
  • There are two flights per day from Newtown to
    Oldtown. The first flight leaves Newtown each day
    at 905 and arrives in Oldtown at 1055. The
    second flight from Newtown leaves at 215. At
    what time does the second flight arrive in
    Oldtown? Show your work.

26
Correct
Incorrect
27
Over- and under-generalization
In which of the following diagrams, is one
quarter of the area shaded?
B
C
D
A
28
Diagnostic item medians
  • What is the median for the following data set?
  • 38 74 22 44 96 22
    19 53
  • 22
  • 38 and 44
  • 41
  • 46
  • 70
  • 77
  • This data set has no median

29
Diagnostic item means
  • What can you say about the means of the following
    two data sets?
  • Set 1 10 12 13 15
  • Set 2 10 12 13 15 0
  • The two sets have the same mean.
  • The two sets have different means.
  • It depends on whether you choose to count the
    zero.

30
Diagnostic item diagonals
Which of the shapes below contains a dotted line
that is also a diagonal?
31
Hinge-point questions
  • A hinge question is based on the important
    concept in a lesson that is critical for students
    to understand before you move on in the lesson.
  • Design requirements
  • Every student must respond to the question within
    two minutes.
  • You must be able to collect and interpret the
    responses from all students in 30 seconds
  • Priorities (in order)
  • In no case should correct and incorrect cognitive
    rules map ontp the correct option
  • Each incorrect option response (distractor)
    should interpret a single cognitive rule
  • Correct option responses (keys) should interpret
    a single cognitive rule

32
Practical techniques feedback
  • Key idea feedback should
  • cause thinking
  • provide guidance on how to improve
  • Comment-only grading
  • Focused grading
  • Explicit reference to rubrics
  • Suggestions on how to improve
  • Not giving complete solutions
  • Re-timing assessment
  • (eg three-quarters-of-the-way-through-a-unit test)

33
Practical techniques sharing learning intentions
  • Explaining learning intentions at start of
    lesson/unit
  • Learning intentions
  • Success criteria
  • Intentions/criteria in students language
  • Posters of key words to talk about learning
  • eg describe, explain, evaluate
  • Planning/writing frames
  • Annotated examples of different standards to
    flesh out assessment rubrics (e.g. reports of
    mathematical investigations)
  • Opportunities for students to design their own
    tests

34
Students owning their learning and as learning
resources
  • Students assessing their own/peers work
  • with rubrics
  • with exemplars
  • two stars and a wish
  • Training students to pose questions/identifying
    group weaknesses
  • Self-assessment of understanding
  • Traffic lights
  • Red/green discs
  • End-of-lesson students review
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